Step By Step Calculus » 17.2 - Integral and Comparison Tests

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Synopsis

There exists a number of other specialized tests to determine whether a series is convergent or divergent. Here are three test mechanisms for nonnegative series.

Integral Test: Suppose \displaystyle \sum_{i=1}^{\infty}a_i\displaystyle \sum_{i=1}^{\infty}a_i is a nonnegative series and f:[0,\infty)\to[0,\infty)f:[0,\infty)\to[0,\infty) is monotonically decreasing with antiderivative FF on [0,\infty)[0,\infty) and satisfies f(i)=a_if(i)=a_i. Then,

\mathbf{p}\mathbf{p}-Series Rule: A pp-series has the form \displaystyle \sum_{i=1}^{\infty}\frac{1}{i^p}\displaystyle \sum_{i=1}^{\infty}\frac{1}{i^p} and is convergent when p>1p>1 and divergent when p\leq 1p\leq 1.